It depends on age and on your systolic pressure.
Diastolic pressure less than 60 mm Hg is generally considered as hypotension,
Severely low blood pressure can deprive the brain and other vital organs of oxygen and nutrients, leading to a shock.
In general, DBP shouldn't get below 35-40 mmHg - this is the filling pressure needed for the coronary arteries to perfuse.
No coronary perfusion, of course, is a very bad thing.
Symptoms may start before this point however, especially if you normally are hypertensive, so I'd start getting concerned once your DBP gets below 55 or so.
Much below 90/60 can be a problem (lead to fainting). But until you're experiencing faintness I'd not worry.